Contents

Idea

A holonomy groupoid is a (topological/Lie-) groupoid naturally associated with a foliationℱ \mathcal{F} of a manifoldX X . It is, in some sense, the smallest de-singularization of the leaf-spacequotientX/ℱ X/\mathcal{F} of the foliation, which is, in general, not a manifold itself. Every foliation groupoid of ℱ \mathcal{F} has this de-singularization property, but the holonomy groupoid is, in some sense, minimal with respect to this property.

Explicitly, given a foliationℱ \mathcal{F} on a manifold X X , the holonomy groupoid of ℱ \mathcal{F} has as objects the points of X X . Given points x,y x,y in the same leaf, a morphism between them is the equivalence class of a path in the leaf from x x to y y , where two paths are identified if they induce the same germ of a holonomy transformation between small transversal sections through x x and y y . If x x and y y are not in the same leaf, then there is no morphism between them.

To a path γ:[0,1]→L \gamma: [0,1] \to L from x x to y y , we assign the germ of a (partially defined) diffeomorphism

holS,T(γ):(S,x)→(T,y),
{hol^{S,T}}(\gamma): (S,x) \to (T,y),

called the holonomy transformation of the path γ \gamma with respect to S S and T T , as follows:

If there exists a single foliation chart U U of ℱ \mathcal{F} that contains the image of γ \gamma , then there exists a sufficiently small open neighborhoodA A of x x in the space S∩U S \cap U for which there exists a unique smooth map f:A→T f: A \to T satisfying the following conditions:

f(x)=y f(x) = y ;

For every a∈A a \in A , the point f(a) f(a) lies in the same plaque in U U as a a . Observe that f f is a diffeomorphism onto its image.

Then define holS,T(γ) {hol^{S,T}}(\gamma) to be the germ of this diffeomorphism at x x :

In general, the image of γ \gamma is not contained inside any single foliation chart U U , but as it is a compact subspace of X X , there exist finitely many foliation charts U1,…,Un+1 U_{1},\ldots,U_{n+1} and numbers t0,…,tn+1 t_{0},\ldots,t_{n+1} such that

Holonomy Groupoid

Given a foliated manifold (X,ℱ) (X,\mathcal{F}) , the monodromy groupoid is the disjoint union of the fundamental groupoids of the leaves of ℱ \mathcal{F} , which is the groupoid having the following properties:

Its objects are the points of X X .

There are no morphisms between two points on different leaves.

The morphisms between two points on the same leaf are homotopy-classes of paths lying in the leaf joining those points.

The holonomy groupoid is defined analogously, where instead of identifying two paths if they are homotopic, they are identified if they induce the same holonomy as described above.